Transport of proteins acrossnanopores: a physicist's perspective
F. Cecconi
CNR-ISC Istituto dei Sistemi Complessi (Roma)
A. Ammenti (Univ. di Perugia, INFN Italy)
U. M.-B.-Marconi (Univ. di Camerino, Italy)
A. Vulpiani (Univ. �“Sapienza�” Roma, INFN, Italy)
M. Chinappi (Univ. �“Sapienza�” Roma, Italy)
M.C. Casciola (Univ. �“Sapienza�” Roma, Italy)
Anomalous Transport: from Billiards to Nanosystems (Sperlonga 2010)
Sperlonga 2010
Overview
�• Biology of translocation (under physics view)
�• Voltage-driven translocation experiments (clean data)
motivation
�• Computer modeling (simulation results)
�• Math. Theory: First Passage Time (interpretation ?)
�• Conclusions
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main �“characters of the tale�” are cellularmembranes and pores which constituteselective biological gates = transport proteins
Translocation (molecular transport)
Proteins (complex biopolymers) areneeded in a variety of locationsinside/outside the cellefficient transport mechanisms
Cell: non isolated chemical lab (continuousexchange of chemical compounds)
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Cell membrane
Barrier separating material inside the cellfrom environment with a complex structure
Structural elements: Lipid bilayer (fatty molecules) +
proteins (receptors,catalyst, mechanical�…�…
transport proteins (pores) allow the exchange of signals, chemicals, genetic information
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Phospholipid bi-layer
Gases (CO2,N2,O2)
Q=0 Small Polar Mol. (Ethanol)
Water & Urea
Q=0 Large Polar Mol. (Glucose)
Ions (K+,Ca2+,Mg2+)
Q 0 Polar Mol.
(Am-Ac,ATP)
Biological relevance of pores (= transport proteins)
Without gates fewmolecules onlywould cross thecell membranes
trappin
g
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-Hemolysin ion channel ( HL)
Transport protein �“mushroom shape�”
2) Can be integrated into lipid bilayers or solid-state substrates to form conducting and transport devices �“nanopore systems�”
Structure resolved bySong et al. Science (1996)
Refined by
Gonaux J.Str.Biol (1998)
Eptameric structure
1) Stable in vitro
Recent interest
with potential technological/biomedical appl.)
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Voltage driven translocation
ion-current drop indicates clogging of the channel single molecule passage
Direct detection of �“translocation�” events
Kasianowicz et al. PNAS (1995ff), Meller et al. PRL (2001)
Experimental setup is a circuit
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Sequencing & mass spectroscopy
the analysis of current signals allows translocatingmolecules to be identified and sequenced, eachmolecule has its own signature in the plane (I,t) ?patterns are affected by unpredictable details of thedynamics (importance of simulations)
Typical current pattern
ideal
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A.Meller et al. PNAS 2000;97:1079-1084
©2000 by The National Academy of Sciences
Sequencing & spectroscopy (II)
IB: Blockage current
tD: translocation time
Well defined clustering
Time and current are sensitive to:length, composition, structure,temperature, channel propertiesit means �“spectroscopy�”!!
characterization of single strandedDNA [poly(dA) and poly(dC)]at different temperatures
STATISTICS
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Common feature: skewnesssimple explanation? Model?
DNA strands
1) No translocation2) Fast translocation3) Slow translocation
MBP Protein
G. Oukhaled et al. PRL 98 (2007)Kasianowicz et al. PNAS 93 (1993)
Distribution of blockage times
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Model (physicist�’s perspective)
Cylindrical channel of finite length L
homogeneous importing force F
Coarse-Grained protein model (united atom) to C -backbone
(also Makarov 2006)
(collecting many events)(preserving UBI structure)
UBI
GS of force field
ubiquitin
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Simplest pore actions
V (ri ) =1 tanh[ x(x L)]
2
yi2
+ zi2
Rp
2
q
Confining effect
Average importing force along x (acting only inside the pore)
Vimp(rk ) = Fxk
L
x
(smooth step)
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Translocation Simulations
Folding equilibrium (setting of parameters)
Runs of translocations to measure
Vel(F), ProbTr(F), time(F) , (t;F) (time distrib.)
At different pulling forces (F)
and temperatures Tref , Tph
Constant temperature MD (Langevin)
Umbrella Sampling + Multiple histograms
mk
d2rk
dt2= m
k r k+ F
kV +R
k
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Examples of Trajectories
analyze trajectories via the reaction(collective) coordinate:center of mass XCM
F=2.5f F=3.0f
Put Ubiquitin near pore entrance and drag it
Pathway: Unfolding transport refolding
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Capture process
monitoring the occupation
of the channel p(tc)
#cis #in #trans
dPc
dt=
c(1 P
c)
pc(t) =dPc
dt= ce
ct
exponential PdF
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Current (Mobility)
=
vix
i=1
N
NF
Non Ohmic behavior of = (F)
indicates the presence of barriers
�“First�” Transport observable
Dominated by fluctuationsinduced by the pore
v = F
TphTref
No Transl
(free) (pore)
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Translocation probability
Given a time window [0,TW], translocation may or
may not occur, thus, we can define aProb. to translocate = #successes/#attempts
Sigmoid shape indicatesCritical Force
Free Energy-Barriers
PTr (Fc) =1 PTr (Fc)
PTr (Fc) =1/2
Fc decreases with Temp.
Tph Tref
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Blockage (translocation) times
Arrhenius-like only on high force regime
(F) = tfp
Aexp( Fa)
Translocation time is given by simulations of firstarrival time at trans side (Mean First Passage Time)
tfp =min{t : XCM(t) L}
cis trans
Random variableTref
Tph
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F =1.38f
Blockage time distribution
first passage time (FPT) at the end of the channel (experiments: time interval of current drop)
not Gaussian with exponential tails controlled by F
F=3.0f
Simple explanation ?
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Free-Energy Profile
G(X) = Gs{1 tanh[
s(X L /2)
2ls
2]}
s=1
3
Transport observables indicated barriers: energylandscape in the reaction coordinate: center of mass X
G(X) = RT lnP(X)
Umbrella simulations restraining the protein in the channel P(X)
Free-en.= U - TS
includes conform.
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Mathematical model (Drift-Diffusion)
Simulation results (exp.) on translocation have a natural interpr. within FPT-problem in terms of a particle undergoing a driven diffusion on a free-energy profile G(X)
J(X,t) = D0eU (X )
Xe
U (X )P(X,t)
U(X) =G(X) FX
+ boundary at channel ends, the most general ones:Radiation BC (Berezhkowskii, Gophic B.J. 2003)
tP = D0 X
eU (X )
Xe
U (X )P
current
L
J(0,t) = R0P(0,t)
J(L,t) = RLP(L,t)
tD
L2
D0
tB
L
0F
tR
L
R
Metzler & Klafter
Anom. Translocation B.J. 2003.
Lubensky & Nelson 99
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First Passage Time theory
Survival prob. S(t) = dX P(X, t)0
L
Blockage Time PdF (t) =dP
out
dt=
dS(t)
dt
(t) = R0P(0,t) + RLP(L, t)
�ˆ (s) = R0�ˆ P (0,s) + R
L�ˆ P (L,s)
Flux at boundaries
Not escaped the channel (time < t)
Recalling Rad-BC
Pout(t) =1 S(t)
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Solving Smoluchovski Equation
Green Function method for s=0 lucky case
PTr (F) = RL
dt P(L,t)0
= RL
�ˆ P (L,0)
(F) = dt dX P(X, t)0
L
0
= dX �ˆ P (X,0)0
L
V (F)L
(F)+ ....
Transloc Prob.
Average time
D0 Xe
U (X )
Xe
U (X )P(X,s){ } sP(X,s) = (X X0)
Laplace Transform �ˆ P (X,s) = dt0
estP(X,t)
Velocity
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Time distribution (unlucky s=0)
0(t;F) =L
4 D0t3exp
[ (F)t L]2
4D0t
t3/2
e(F )t /(4D0 )
strategy to improve the fitting to data
�ˆ (s) = R0�ˆ P (0,s) + R
L�ˆ P (L,s)
Absorption0 L
source
the shape of PdF is reasonable, but the fitting of tails can be often unsatisfactory, can we do better?
Inverse Gaussian
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Re-absorbe approximation into BC (mimicking thepresence of barriers at the channel ends)
J = FP D0
P
xP
G
x
J(0,t) = [R0 + R0 ]P(0,t) R0 G'(0)
J(L,t) = [RL+ R
L]P(L, t) R
LG'(L)
now an analytical expressionof Laplace transf. can be derived
Simplest driftdiffusion eq.
�“Dirty trick�” attempt
XJ(X,s){ } sP(X,s) = (X X0 )
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) (s;F) =
RZ(s)evL /2D0
RZ(s)cosh( L)+ (vR + 2D0s)sin( L)
Z(s) = v2 + 4D0s = Z(s) /2D0 v = F
RBC(t;F) = a(t)exp( t)
only numerical inversion: large timebehaviour is controlled by the 1st Pole
helps the fitting procedure: constrain
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Conclusions
C.G. Model of Protein Translocation reproduces basic
phenomenology.
Support to 1dim Driven-Diffusion model: D0, 0,RL Free-energy profile G(X)=-RT log P(X) MESSAGE
Driven-Diffusion and FTP theory work: conceptualflexible framework to interpret translocation data:a step farther than mere fitting.
Limitation: reliable reaction coordinates and
some knowledge on free-energy landscape
(Ammenti et al. JPCB 113, 10348 (2009))
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Go model: outline
Folding ruled by topology of native state
Force Field: Native Structure bias
No need of sequence = Ideal Sequence
(Minimal Frustration)
Support: Folding time vs contact order
K.W. Plaxco et al. (1998) JMB
Go-model confers protein-like properties to chains
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Go-model for Ubiquitin
Dihedral potential (torsion)
V = k(1)[1 cos( )]+
k(3)[1 cos3( )]
V ( ) =k
2( )
2
Bending potential (elasticity) Chain potential
Backbone of C carbon (Native interactions)
Preserves Protein-like properties
V (r) =kh
2(ri,i+1 R
i,i+1)2
Clementi et al. JMB 2000
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Given a Protein Native state
Choose distance cutoff Rc and
Define Native Interactions Rij < Rc
Vnat = 5Rij
rij
12
6Rij
rij
10
i, j> i+1
Vnnat =10
3i, j> i+1
rij
12
PDB structure is the ground state of Go Force-Field